Electron localization in exact time-dependent density-functional potentials

By propagation of the exact many-electron wavefunction, we determine exact Kohn-Sham (KS) potentials for 1D systems with strong correlation [1]. From this we have developed a density functional which incorporates several features, present in the exact KS potential, that are entirely missing from the usual approximations made in time-dependent density-functional theory (TDDFT) [2]. We find a strong and time-dependent self-interaction correction, owing to electron localization, as well as large static and dynamic spatial steps in the KS potential. Our new functional, suited to simulating ground-state and time-dependent electronic systems, combines an expression for the exact KS potential in the limit of complete electron localization with a measure of the actual localization. Self-consistent application of the functional provides accurate densities for a range of systems, even where the KS potential requires non-local dependence on the charge density, such as potential steps; the self-interaction correction is accurately described. We explore the relationship between features in the KS potential and the ``derivative discontinuity.'' \\[4pt] [1] M. J. P. Hodgson \textit{et al.} \textbf{Phys. Rev. B} 88, 241102(R)\\[0pt] [2] M. J. P. Hodgson \textit{et al.} \textbf{arXiv:1409.5666}